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Boundary maps vanish for a split tower
Statement
Suppose a decreasing augmented tower is equipped with an equivalence of augmented towers to the direct-sum tail tower of Graded pieces of a direct-sum tail tower. Then every connecting map is zero. In particular, every connecting homomorphism on homotopy groups in the stage long exact sequences is zero.
Facts & Assumptions
Given: The augmented tower and the specified equivalence of towers.
Goal: Prove the vanishing of its stage boundaries.
Decreasing spectral filtrations and associated graded defines the graded pieces as cofibers of transitions.
Graded pieces of a direct-sum tail tower gives zero connecting maps for the tail tower.
Proof
The equivalence of tower diagrams identifies each transition map with the corresponding tail inclusion. Functoriality of cofiber sequences identifies the connecting map of the given transition with that of the tail inclusion.
The latter map is zero by [F2], including in negative filtration degrees. Applying the homotopy-group functor sends a zero map to the zero homomorphism, which proves both assertions.
Depends on
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